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Properties

Label 171.4.f.h
Level $171$
Weight $4$
Character orbit 171.f
Analytic conductor $10.089$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,4,Mod(64,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.f (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.0893266110\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 50x^{10} + 1797x^{8} + 29198x^{6} + 345409x^{4} + 2092128x^{2} + 8856576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} + 9 \beta_{2} - 9) q^{4} + (\beta_{10} + \beta_1) q^{5} + (\beta_{11} - \beta_{4} - 5) q^{7} + ( - \beta_{8} + \beta_{5} - 6 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} + 9 \beta_{2} - 9) q^{4} + (\beta_{10} + \beta_1) q^{5} + (\beta_{11} - \beta_{4} - 5) q^{7} + ( - \beta_{8} + \beta_{5} - 6 \beta_{3}) q^{8} + ( - 5 \beta_{11} + 5 \beta_{9} + \cdots - 8) q^{10}+ \cdots + ( - 40 \beta_{10} + 10 \beta_{5} - 238 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 52 q^{4} - 60 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 52 q^{4} - 60 q^{7} - 56 q^{10} + 178 q^{13} - 172 q^{16} + 296 q^{19} - 944 q^{22} - 458 q^{25} - 140 q^{28} + 196 q^{31} - 400 q^{34} + 628 q^{37} + 48 q^{40} + 642 q^{43} - 960 q^{46} - 2296 q^{49} - 300 q^{52} + 648 q^{55} - 2880 q^{58} + 3446 q^{61} + 9528 q^{64} + 2730 q^{67} - 1920 q^{70} + 1690 q^{73} - 4308 q^{76} - 2898 q^{79} - 1616 q^{82} - 4176 q^{85} + 11856 q^{88} - 3210 q^{91} - 5280 q^{94} - 1180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 50x^{10} + 1797x^{8} + 29198x^{6} + 345409x^{4} + 2092128x^{2} + 8856576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 421097 \nu^{10} + 18574850 \nu^{8} + 667580109 \nu^{6} + 9091814878 \nu^{4} + \cdots + 777219275616 ) / 511964824416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 421097 \nu^{11} - 18574850 \nu^{9} - 667580109 \nu^{7} - 9091814878 \nu^{5} + \cdots - 265254451200 \nu ) / 511964824416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2500\nu^{10} - 89850\nu^{8} - 3229209\nu^{6} - 17270450\nu^{4} - 104606400\nu^{2} + 5014036725 ) / 516093573 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2500\nu^{11} - 89850\nu^{9} - 3229209\nu^{7} - 17270450\nu^{5} - 104606400\nu^{3} + 7594504590\nu ) / 516093573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4678649 \nu^{10} - 226641250 \nu^{8} - 8145486525 \nu^{6} - 137428566526 \nu^{4} + \cdots - 9483250101600 ) / 511964824416 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28392391 \nu^{11} + 2062931550 \nu^{9} + 74141759907 \nu^{7} + 1557563627618 \nu^{5} + \cdots + 86318337156768 \nu ) / 3071788946496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3392067 \nu^{11} + 159757750 \nu^{9} + 5741693535 \nu^{7} + 91443820458 \nu^{5} + \cdots + 6684673239840 \nu ) / 255982412208 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28392391 \nu^{10} + 2062931550 \nu^{8} + 74141759907 \nu^{6} + 1557563627618 \nu^{4} + \cdots + 86318337156768 ) / 1535894473248 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 64850 \nu^{11} - 2330709 \nu^{9} - 73443810 \nu^{7} - 447995473 \nu^{5} + \cdots + 28135631979 \nu ) / 3096561438 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64850 \nu^{10} - 2330709 \nu^{8} - 73443810 \nu^{6} - 447995473 \nu^{4} - 2713490016 \nu^{2} + 28135631979 ) / 1548280719 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 17\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{5} - 22\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} - 40\beta_{6} - 377\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40\beta_{8} - 6\beta_{7} + 577\beta_{3} - 577\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 150\beta_{11} - 1297\beta_{4} + 9875 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 300\beta_{10} - 1297\beta_{5} + 16360\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -5391\beta_{11} + 5391\beta_{9} + 39706\beta_{6} + 39706\beta_{4} + 279311\beta_{2} - 279311 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -10782\beta_{10} - 39706\beta_{8} + 10782\beta_{7} + 39706\beta_{5} - 477841\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -173028\beta_{9} - 1192549\beta_{6} - 8145377\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1192549\beta_{8} - 346056\beta_{7} + 14108122\beta_{3} - 14108122\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−2.72529 − 4.72034i
−1.71809 − 2.97581i
−1.45636 − 2.52249i
1.45636 + 2.52249i
1.71809 + 2.97581i
2.72529 + 4.72034i
−2.72529 + 4.72034i
−1.71809 + 2.97581i
−1.45636 + 2.52249i
1.45636 − 2.52249i
1.71809 − 2.97581i
2.72529 − 4.72034i
−2.72529 − 4.72034i 0 −10.8544 + 18.8003i −1.48079 − 2.56480i 0 6.79546 74.7206 0 −8.07113 + 13.9796i
64.2 −1.71809 − 2.97581i 0 −1.90364 + 3.29720i 6.83777 + 11.8434i 0 −20.1525 −14.4069 0 23.4957 − 40.6958i
64.3 −1.45636 − 2.52249i 0 −0.241981 + 0.419123i −10.1021 − 17.4973i 0 −1.64299 −21.8921 0 −29.4246 + 50.9649i
64.4 1.45636 + 2.52249i 0 −0.241981 + 0.419123i 10.1021 + 17.4973i 0 −1.64299 21.8921 0 −29.4246 + 50.9649i
64.5 1.71809 + 2.97581i 0 −1.90364 + 3.29720i −6.83777 − 11.8434i 0 −20.1525 14.4069 0 23.4957 − 40.6958i
64.6 2.72529 + 4.72034i 0 −10.8544 + 18.8003i 1.48079 + 2.56480i 0 6.79546 −74.7206 0 −8.07113 + 13.9796i
163.1 −2.72529 + 4.72034i 0 −10.8544 − 18.8003i −1.48079 + 2.56480i 0 6.79546 74.7206 0 −8.07113 − 13.9796i
163.2 −1.71809 + 2.97581i 0 −1.90364 − 3.29720i 6.83777 − 11.8434i 0 −20.1525 −14.4069 0 23.4957 + 40.6958i
163.3 −1.45636 + 2.52249i 0 −0.241981 − 0.419123i −10.1021 + 17.4973i 0 −1.64299 −21.8921 0 −29.4246 − 50.9649i
163.4 1.45636 − 2.52249i 0 −0.241981 − 0.419123i 10.1021 − 17.4973i 0 −1.64299 21.8921 0 −29.4246 − 50.9649i
163.5 1.71809 − 2.97581i 0 −1.90364 − 3.29720i −6.83777 + 11.8434i 0 −20.1525 14.4069 0 23.4957 + 40.6958i
163.6 2.72529 − 4.72034i 0 −10.8544 − 18.8003i 1.48079 − 2.56480i 0 6.79546 −74.7206 0 −8.07113 − 13.9796i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.f.h ✓ 12
3.b odd 2 1 inner 171.4.f.h ✓ 12
19.c even 3 1 inner 171.4.f.h ✓ 12
57.h odd 6 1 inner 171.4.f.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.4.f.h ✓ 12 1.a even 1 1 trivial
171.4.f.h ✓ 12 3.b odd 2 1 inner
171.4.f.h ✓ 12 19.c even 3 1 inner
171.4.f.h ✓ 12 57.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{12} + 50T_{2}^{10} + 1797T_{2}^{8} + 29198T_{2}^{6} + 345409T_{2}^{4} + 2092128T_{2}^{2} + 8856576 \) Copy content Toggle raw display
\( T_{7}^{3} + 15T_{7}^{2} - 115T_{7} - 225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 50 T^{10} + \cdots + 8856576 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 448364160000 \) Copy content Toggle raw display
$7$ \( (T^{3} + 15 T^{2} + \cdots - 225)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} - 5044 T^{4} + \cdots - 488138400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 89 T^{5} + \cdots + 3005780625)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 98\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} - 148 T^{5} + \cdots + 322687697779)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} - 49 T^{2} + \cdots - 2385261)^{4} \) Copy content Toggle raw display
$37$ \( (T^{3} - 157 T^{2} + \cdots + 28025385)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 11\!\cdots\!25)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 20\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 25\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 20\!\cdots\!25)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 65\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 1811716000000)^{2} \) Copy content Toggle raw display
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